On the variety of tritangential planes to a general K3 surface of degree 6 and genus 4 in ${\text{mP}}^4$

Let $S\subset {\text{mP}}^4$ be a general K3 surface of degree 6 and genus 4. In this paper we study the irreducible variety $X_S$ of \emph{tritangential planes} to $S$ whose general point is a plane that intersects $S$ in a curvilinear scheme of length six supported at three non collinear points. The variety $X_S$ can be identified as the relevant part of the fixed locus of the so called \emph{Beauville involution} defined on the Hilbert scheme $S[3]$ of 0–dimensional schemes of length three of $S$. In this paper we prove that: (a) $X_S$ has dimension 3, is irreducible and smooth, except for 210 points that are at most of multiplicity 2 for $X_S$; (b) $X_S$, in its natural embedding in the Grassmannian $\mathbb{G}(2,4)\subset {\text{mP}}^9$ of planes in ${\text{mP}}^4$, has degree $152$.